Evoked Potentials (EPs)

Transcription

1 EVOKED POTENTIALS

2 Evoked Potentials (EPs) Event-related brain activity where the stimulus is usually of sensory origin. Acquired with conventional EEG electrodes. Time-synchronized = time interval from stimulus to response is usually constant.

3 EP = A Transient Waveform Evoked potentials are usually hidden" in the EEG signal. Their amplitude ranges from µv, to be compared with µv of the EEG. Their duration is milliseconds.

4 Examples of Evoked Potentials Note the widely different amplitudes and time scales.

5 EP Definitions Amplitude Time for stimulus Latency

6 Auditive Evoked Potentials AEPs

7 Visual Evoked Potentials VEPs

8 Somatosensory Evoked Potentials SEPs

9 SEPs during Spinal Surgery stimulation recording electrodes electrode #1 electrode #2

10 EP Scalp Distribution

11 A. Evoked potentials resulting from a color task in which red and blue flashed checkerboards were presented in a rapid, randomized sequence at the center of the screen. B. Scalp voltage distributions evoked potentials at different latency ranges.

12 Brainstem Auditive EP (BAEPs) in Newborns wave III stimulus 2 months I III IV V 1 ms 8 months I II III IV V VI VII

13 BAEPs of Healthy Children 2 8 years 0 1 years wave I II III IV V newborn premature latency (ms)

14 Cognitive EPs

15 Ensemble Formation

16 Formation of an EP Ensemble stimulus# EEG signal EP ensemble

17 10 Superimposed EPs amplitude (microv) latency (ms)

18 Model for Ensemble Averaging fixed shape

19 Noise Assumptions I. II. III.

20 Ensemble Averaging The ensemble average is defined by The more familiar (scalar) expression for ensemble averaging is given by

21 evoked potentials Ensemble Averaging

22 Noise Variance The variance of the ensemble average is inversely proportional to the the number of averaged potentials, that is:

23 Reduction of Noise Level The noise estimate before division by the reduction factor 1/ M Reduction in noise level of the ensemble average as a function of #potentials #potentials

24 Exponential averaging The ensemble average can be computed recursively because: assuming Exponential averaging results from replacing the weight 1/M with alpha:

25 Exponential averaging

26 Noise Reduction of EPs with Varying Noise Level Assumption: all evoked potentials have identical shapes s(n) but with varying noise level. Such an heterogenous ensemble is processed by weighted averaging.

27 Weighted Averaging The weighted average is obtained by weighting each potential x i (n) with its inverse noise variance: where each weight w i thus is This expression reduces to the ensemble average when the noise variance is identical in all potentials.

28 Weighted Averaging, cont How to estimate the varying noise level?

29 Weighted averaging: An Example W eight ed a vera ge The ensemble consists of 80 EPs with variance 1 and 20 EPs with variance 20 (heterogenous) Ensemble a vera ge

30 Robust Waveform Averaging Gaussian noise Laplacian noise

31 The Effect of Latency Variations Signal model: x i (n)=s(n θ i )+v i (n)

32 Lowpass Filtering of the Signal The expected value of the ensemble average, in the presence of latency variations, is given by: or, equivalently, in the frequency domain:

33 Latency Variation and Lowpass Filtering Gaussian PDF Uniform PDF

34 Techniques for Correction of Latency Variations Synchronize with respect to a peak of the signal or similar property. Crosscorrelation between two EPs. Woody s method for iterative synchronization of all responses of the ensemble. The method terminates when no further latency corrections are done.

35 Estimation of Latency An Illustration Input signal Template waveform Correlation function Latency estimate

36 Woody s Method

37 Woody s Method: Different SNRs good SNR not so good SNR bad SNR

38 SNR-based Weighting Design a weight function w(n) which minimizes E [ (s(n) ŝ a (n)w(n)) 2] where s(n) denotes the desired signal and ŝ a (n) the ensemble average. The optimal filter is w(n)= σ2 s(n) σ 2 s(n)+ σ2 v M = σ2 v Mσ 2 s(n)

39 SNR-based Weighting Noise-free signal Ensemble average Weight function Weight function multiplied with ensemble average

40 Noise Reduction by Filtering Estimate the signal and noise power spectra from the ensemble of signals. Design a linear, time-invariant, linear filter such that the mean square error is minimized, i.e., design a Wiener filter. Apply the Wiener filter to the ensemble average to improve its SNR.

41 Wiener Filtering S s (e jω ) S v (e jω ) : signal power spectrum : noise power spectrum Wiener filter: jω H(e jω S s (e jω ) )= S S s (e jω )+ 1S v M S (e jω ) s (e jω )+ v(e jω ) for one potential for M potentials

42 increasing SNR Filtering of Evoked Potentials

43 Limitations of Wiener filtering Assumes that the observed signal is stationary (which in practice it is not...). Filtering causes the EP peak amplitudes to be severely underestimated at low SNRs. As a result, this technique is rarely used in practice.

44 Tracking of EP Morphology So far, noise reduction has been based on the entire ensemble, e.g., weighted or exponential averaging We will now track changes in EP morphology by socalled single-sweep analysis. More a priori information is introduced by describing each EP by a set of basis functions.

45 Selection of Basis Functions Orthonormality is an important function property of basis functions. Sines/cosines are well-known basis functions, but it is often better to use......functions especially determined for optimal (MSE) representation of different waveform morphologies (the Karhunen-Loève representation).

46 Orthogonal Expansions

47 Basis Functions: An Example Linear combinations of two basis functions model a variety of signal morphologies

48 Calculation of the Weights

49 Mean-Square Weight Estimation i.e. identical to the previous expression

50 Truncated Expansion The underlying idea of signal estimation through a truncated series expansion is that a subset of basis functions can provide an adequate representation of the signal part. Decomposition into signal and noise parts: The estimate of the signal is obtained from:

51 Truncated Expansion, cont

52 Examples of Basis Functions Sine/ Cosine Walsh

53 Sine/Cosine Modeling #basis functions K = 3 K = 7 K = 12 K = 500 VEP without noise

54 Sine/Cosine Modeling: Amplitude Estimate and MSE Error

55 MSE Basis Functions How should the basis functions be designed so that the signal part is efficiently represented with a small number of functions? We start our derivation by decomposing the series expansion of the signal into two sums, that is,

56 Karhunen Loève Basis Functions The Karhunene Loève (KL) basis functions, minimizing the MSE, are obtained as the solution of the ordinary eigenvalue problem, and equals the eigenvectors corresponding to the largest eigenvalues: The MSE equals the sum of the (N-K) smallest eigenvalues

57 KL Performance Index Example of the performance index

58 How to get Rx?

59 Example: KL Basis Functions Basis functions Signals Observed signal: x i Signal estimate: ŝ i

60 Time-Varying Filter Interpretation

61 Modeling with Damped Sinusoids The original Prony method The least-squares Prony method Variations

62 Adaptive Estimation of Weights

63 Adaptive Estimation of Weights The instantaneous LMS algorithm, in which the weights of the series expansion are adapted at every time instant, thereby producing a weight vector w(n) The block LMS algorithm, in which the weights are adapted only once for each EP ( block ), thereby producing a weight vector w i that corresponds to the i:th potential.

64 Estimation Using Sine/Cosine

65 Estimation Using KL Functions

66 Limitations Sines/cosines and the KL basis functions lack the flexibility to efficiently track changes in latency of evoked potentials, i.e., changes in waveform width. The KL basis functions are not associated with any algorithm for fast computations since the functions are signal-dependent.

67 Wavelet Analysis Wavelets is a very general and powerful class of basis functions which involve two parameters: one for translation in time and another for scaling in time. The purpose is to characterize the signal with good localization in both time and frequency. These two operations makes it possible to analyze the joint presence of global waveforms ( large scale ) as well as fine structures ( small scale ) in a signal. Signals analyzed at different scales, with an increasing level of detail resolution, is referred to as a multiresolution analysis.

68 Wavelet Applications signal characterization signal denoising data compression detecting transient waveforms and much more!

69 The Correlation Operation Recall the fundamental operation in orthonormal basis function analysis: in discrete-time, the correlation between the observed signal x(n) and the basis functions ϕ k (n): In wavelet analysis, the two operations of scaling and translation in time are most simply introduced when the continuous-time description is adopted:

70 The Mother Wavelet

71 The Wavelet Transform C W T I C W T

72 The Scalogram Composite signal Scalogram

73 The Discrete Wavelet Transform The CWT w(s, τ ) is highly redundant and needs to be sampled Dyadic sampling The discretized wavelet function The discrete wavelet transform (DWT) The inverse discrete wavelet transform (IDWT)

74 Multiresolution Analysis A signal can be viewed as the sum of a smooth ( coarse ) part and a detailed ( fine ) part. The smooth part reflects the main features of the signal, therefore called the approximation signal. The faster fluctuations represent the signal details. The separation of a signal into two parts is determined by the resolution with which the signal is analyzed, i.e., by the scale below which no details can be discerned.

75 Multiresolution Analysis Exemplified

76 Multiresolution Analysis, cont In mathematical terms this is expressed as:

77 The Scaling Function The scaling function ϕ(t) is introduced for the purpose of efficiently representing the approximation signal x j (t) at different resolution. This function, being related to a unique wavelet function ψ(t), can be used to generate a set of scaling functions defined by different translations: where the index 0 indicates that no time scaling is performed.

78 The Scaling Function, cont The design of a scaling function ϕ(t) must be such that translations of ϕ(t) constitute an orthonormal set of functions, i.e., Its design is not considered in this course, but some existing scaling functions are applied.

79 The Approximation Signal x0(t)

80 The Approximation Signal xj(t) (dyadic sampling)

81 The Multiresolution Property

82 The Refinement Equation h ϕ (n) is a sequence of scaling coefficients

83 The Wavelet Function It is desirable to introduce the function ψ(t) which complements the scaling function by accounting for the details of a signal rather than its approximations. For this purpose, a set of orthonormal basis functions at scale j is given by which spans the difference between the two subspaces Vj and Vj+1.

84 Scaling and Wavelet Functions

85 Orthogonal Complements

86 The Wavelet Series Expansion Compare this expansion with the orthogonal expansions mentioned earlier such as the one with sine/cosine basis functions, i.e., the Fourier series. The wavelet/scaling coefficients do not have a similar simple interpretation.

87 Multiresolution Signal Analysis: A Classical Example The Haar scaling function The Haar wavelet function These functions are individually and mutually orthonormal

88 The Haar Scaling Function

89 Haar Multiresolution Analysis Approximation signals Detail signals

90 Haar Scaling and Wavelet Functions

91 Computation of Coefficients The scaling and wavelet coefficients can computed recursively by exploring the refinement equation so that, for example, the scaling coefficients are computed with see derivation on page 300

92 Filter Bank Implementation

93 DWT Calculation

94 Inverse DWT Calculation

95 Scaling Function Examples

96 Coiflet Multiresolution Analysis

97 Scaling Coefficients in Noise

98 Denoising of Evoked Potentials

99 EP Wavelet Analysis Visual EP coefficients of W3 reconstructed waveform coefficients of V3 reconstructed waveform from Ademoglu et al., 1997

100 EP Wavelet Analysis, cont Waveforms reconstructed from V3 and superimposed for 24 normal subjects (upper panel) and for 16 patients with dementia (lower panel). Normal Dement

101 EMG IS NOT COVERED IN THIS COURSE